Resonator element and resonator having a tapered arm next to the base

ABSTRACT

A resonator element includes: at least one resonating arm which performs flexural vibration; a base portion connected to an end of the resonating arm; and a tapered portion which is axisymmetrical with respect to a centerline which bisects the width of the resonating arm, and which has a width increasing toward a portion of the tapered portion connected to the base portion from a portion of the tapered portion connected to the resonating arm, wherein assuming that the length and width of the resonating arm are L and W and the length and width of the tapered portion are Lt and Wt, the shape of the tapered portion is controlled to satisfy a taper length occupancy η=Lt/L and a taper width occupancy ξ=2 Wt/W.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of U.S. patentapplication Ser. No. 14/255,421 filed Apr. 17, 2014, which is acontinuation application of U.S. patent application Ser. No. 13/079,163filed on Apr. 4, 2011, which claims the benefit of Japanese PatentApplication No. 2010-089259 filed Apr. 8, 2010, all of which areincorporated herein by reference in their entireties.

BACKGROUND

1. Technical Field

The present invention relates to a resonator element and a resonatorhaving the resonator element.

2. Related Art

There is known a resonator element which has a resonating arm and a baseportion connected to the resonating arm and in which a tapered portionsymmetrical with respect to the vibration center of the resonating armis provided in a connection portion between the resonating arm and thebase portion (for example, refer to JP-A-2005-5896).

In the resonator element disclosed in JP-A-2005-5896, by providing atapered portion in a connection portion between the resonating arm andthe base portion, leakage of fundamental vibration is suppressed, the Qfactor is increased accordingly so that a vibration mode is notaffected, and it is also possible to prevent the resonator element frombeing broken by an impact or prevent deterioration of a vibration mode.

In the case where a tapered portion is provided in practice, however,the Q factor is often decreased depending on the shape of the taperedportion. Presumably, this is because a thermoelastic loss resulting fromthe shape of the tapered portion exceeds an increase in the Q factorresulting from a suppression effect of vibration leakage by providingthe tapered portion.

SUMMARY

An advantage of the invention is to solve at least a part of theabove-described problems, and the invention can be implemented as thefollowing forms or application examples.

APPLICATION EXAMPLE 1

According to this application example of the invention, there isprovided a resonator element including: at least one resonating armwhich performs flexural vibration; a base portion connected to an end ofthe resonating arm; and a tapered portion which is axisymmetrical withrespect to the vibration center of the resonating arm and which has awidth increasing toward a portion of the tapered portion connected tothe base portion from a portion of the tapered portion connected to theresonating arm. Assuming that length and width of the resonating arm areL and W and length and width of the tapered portion are Lt and Wt, theshape of the tapered portion is controlled to satisfy a taper lengthoccupancy η=Lt/L and a taper width occupancy ξ=2Wt/W.

According to this application example of the invention, by controllingthe shape of the tapered portion with the taper length occupancy η andthe taper width occupancy ξ, it is possible to find out the shape of thetapered portion capable of suppressing the thermoelastic loss. As aresult, it is possible to realize a resonator element with a stablevibration mode by increasing the Q factor compared with the case where atapered portion is not provided or the case where a tapered portion issimply provided.

APPLICATION EXAMPLE 2

In the resonator element according to the above application example ofthe invention, it is preferable that the taper width occupancy ξ is in arange of2.409η²+4.728×10⁻²η+2.959×10⁻²≦ξ≦−4.723×10¹η²+1.434×10¹η+1.711×10⁻² whenthe range of the taper length occupancy η is 0.016≦η≦0.1680 and that theshape of the tapered portion is formed by a straight line or a curvedline connecting the position of the tapered portion connected to theresonating arm and the position thereof connected to the base portion.

By setting the taper width occupancy ξ within such a range with respectto the taper length occupancy η, it is possible to obtain a Q factorabout 1.1 times the Q factor when the tapered portion is not provided.

APPLICATION EXAMPLE 3

In the resonator element according to the above application example ofthe invention, it is preferable that the taper width occupancy is in arange of9.417η²−6.358×10⁻¹η+9.984×10⁻²≦ξ≦−5.255×10¹η²+1.394×10¹η−1.219×10⁻¹ whenthe range of the taper length occupancy η is 0.0168≦η≦0.1680 and thatthe shape of the tapered portion is formed by a straight line or acurved line connecting the position the tapered portion connected to theresonating arm and the position thereof connected to the base portion.

By setting the taper width occupancy ξ within such a range with respectto the taper length occupancy η, it is possible to obtain a Q factorabout 1.2 times the Q factor when the tapered portion is not provided.

APPLICATION EXAMPLE 4

In the resonator element according to the above application example ofthe invention, it is preferable that the taper width occupancy ξ is in arange of5.191×10¹η²−6.959η+4.339×10⁻¹≦ξ≦−9.160×10¹η²+1.899×10¹η−4.679×10⁻¹ whenthe range of the taper length occupancy η is 0.0504≦η≦0.1260 and thatthe shape of the tapered portion is formed by a straight line or acurved line connecting the position of the tapered portion connected tothe resonating arm and the position thereof connected to the baseportion.

By setting the taper width occupancy ξ within such a range with respectto the taper length occupancy η, it is possible to obtain a Q factorabout 1.3 times the Q factor when the tapered portion is not provided.

APPLICATION EXAMPLE 5

In the resonator element according to the above application example ofthe invention, it is preferable that the shape of the tapered portion isformed by connection of a plurality of straight lines and that crossingangles between the plurality of straight lines are smaller than acrossing angle between an extended line of the resonating arm and astraight line connecting the position of the tapered portion connectedto the resonating arm and the position thereof connected to the baseportion.

Thus, if the tapered portion has a shape formed by connection of aplurality of straight lines, stress generated in the tapered portion isdispersed. Accordingly, since distortion is also dispersed, temperaturegradient is decreased and the heat flow is suppressed. As a result, athermoelastic loss is reduced and the Q factor can be increased.

APPLICATION EXAMPLE 6

In the resonator element according to the above application example ofthe invention, it is preferable that the shape of the tapered portion isformed by one continuous curved line connecting the position of thetapered portion connected to the resonating arm and the position thereofconnected to the base portion and that a crossing angle between thecurved line and the resonating arm is smaller than a crossing anglebetween an extended line of the resonating arm and a straight lineconnecting the position of the tapered portion connected to theresonating arm and the position thereof connected to the base portion.

As described above, since stress and distortion can be dispersed and thethermoelastic loss can be reduced by forming the shape of a taperedportion by a plurality of straight lines, as the number of the straightlines increases, the thermoelastic loss can be reduced accordingly.Therefore, if the shape of the tapered portion is formed by a curvedline, the thermoelastic loss can be further reduced.

APPLICATION EXAMPLE 7

According to this application example of the invention, there isprovided a resonator including the resonator element described in any ofthe above examples, which is housed in a package.

The resonator element described above is housed in a package formed ofceramic, for example. The inside of the package is preferably in avacuum state. When the resonator element vibrates in a vacuumenvironment, it can maintain more stable vibration over a long period oftime.

In addition, since the resonator element is housed in the package, itshandling is easy and it can be protected against external environments,such as humidity.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be described with reference to the accompanyingdrawings, wherein like numbers reference like elements.

FIG. 1 is a plan view showing the schematic configuration of a resonatorelement according to a first embodiment.

FIG. 2A is a cross-sectional view taken along the line A-A of FIG. 1.

FIG. 2B is a partial plan view showing a B portion of FIG. 1 in anenlarged manner.

FIG. 3 is a graph showing the relationship among the taper lengthoccupancy η, the taper width occupancy ξ, and a Q factor.

FIG. 4 is a graph showing the relationship between the taper lengthoccupancy η (horizontal axis) at which Q/Q₀ is equal to or larger than1.1 and the taper width occupancy ξ (vertical axis).

FIG. 5 is a graph showing the relationship between the taper lengthoccupancy η (horizontal axis) at which Q/Q₀ is equal to or larger than1.2 and the taper width occupancy ξ (vertical axis).

FIG. 6 is a graph showing the relationship between the taper lengthoccupancy η (horizontal axis) at which Q/Q₀ is equal to or larger than1.3 and the taper width occupancy ξ (vertical axis).

FIG. 7 is a partial plan view showing a tapered portion related to afourth example.

FIG. 8 is a partial plan view showing a tapered portion related to afifth example.

FIG. 9 is a partial plan view showing a tapered portion related to asixth example.

FIGS. 10A and 10B are views showing the schematic configuration of aresonator, where FIG. 10A is a plan view and FIG. 10B is across-sectional view taken along the line E-E of FIG. 10A.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

Hereinafter, embodiments of the invention will be described withreference to the accompanying drawings.

Moreover, in the drawings referred to in the following explanation, thescale of each member or portion is adjusted for convenience ofillustration.

First Embodiment

FIG. 1 is a plan view showing the schematic configuration of a resonatorelement according to a first embodiment. FIG. 2A is a cross-sectionalview taken along the line A-A of FIG. 1, and FIG. 2B is a partial planview showing a B portion of FIG. 1 in an enlarged manner. First, theschematic configuration of a resonator element will be described withreference to FIG. 1. A resonator element 10 is spread out on the planehaving an X axis and a Y axis perpendicular to the X axis, and a Z axisindicates a thickness.

Although a material of the resonator element 10 according to the presentembodiment is not particularly limited, it may be quartz crystal, apiezoelectric material other than quartz crystal, or other materials. Inthe case of quartz crystal, it is cut out from a single crystal ofquartz crystal such that the X, Y, and Z axes become an electrical axis,a mechanical axis, and an optical axis, respectively.

In addition, although the resonator element 10 according to the presentembodiment shows an element used for a resonator, it may also be appliedto an acceleration sensor or a gyro sensor.

The resonator element 10 includes: two parallel resonating arms 30 whichextend in the Y-axis direction and flexurally vibrate in the X-axisdirection; a base portion 20 connected to one end of the respectiveresonating arms 30; a spindle portion 40 provided at the other end; andtwo support arms 50 which protrude from the end surface facing a portionof the base portion 20 connected to the resonating arms 30 and whichextend in parallel at both sides of the resonating arms 30.

A groove 35 is formed in the resonating arm 30. As shown in FIG. 2A, thegroove 35 is formed at the same position on both the top and bottomsurfaces of each of the two resonating arms 30 so as to have the sameposition, the same width, the same length, and the same depth, and hasan approximately H sectional shape.

In addition, tapered portions 25 are formed in the resonating arm 30 inthe ±X directions of the resonating arm 30. The tapered portions 25 aresymmetrical to each other with respect to the vibration center of theresonating arm 30 (axisymmetrical with respect to the centerline whichbisects the width of the resonating arm 30). In addition, the taperedportion 25 has a shape whose width increases toward a portion of thetapered portion 25 which is connected to the base portion 20, from aportion of the tapered portion 25 connected to the resonating arm 30.The shape of the tapered portion 25 will be described with reference toFIG. 2B.

Here, the width and length of the tapered portion are expressed as Wtand Lt, and the width and length of the resonating arm 30 are expressedas W and L. In addition, the length L of the resonating arm 30 is alength including the spindle portion 40 and the tapered portion 25. Inaddition, the ratio of the width Wt of the tapered portion 25 and thewidth W of the resonating arm 30 is expressed as a taper width occupancyξ, and the ratio of the length Lt of the tapered portion 25 and thelength L of the resonating arm 30 is expressed as a taper lengthoccupancy η. Therefore, the taper width occupancy can be expressed asξ=2Wt/W, and the taper length occupancy can be expressed as η=Lt/L.

In the present embodiment, it was found out that thermoelastic loss canbe suppressed by controlling the shape of the tapered portion 25 withthe taper length occupancy η and the taper width occupancy ξ. This willbe described with reference to the accompanying drawings.

FIG. 3 is a graph showing the relationship among the taper lengthoccupancy η, the taper width occupancy ξ, and a Q factor. FIG. 3 shows achange of Q/Q₀ (vertical axis), which is a standardized Q factor, withrespect to the taper width occupancy ξ (horizontal axis) when the taperlength occupancy η is made to change from η=0.0168 to η=0.1680 as aparameter. In addition, Q/Q₀ is a standardized ratio of Q₀, obtained bysettling a Q factor when there is no tapered portion, and Q, which is aQ factor when there is a tapered portion.

As shown in FIG. 3, in the taper length occupancy η range of 0.0168 to0.1680, Q/Q₀ is equal to or larger than 1 in a certain range of thetaper width occupancy ξ. That is, there is a combination in which the Qfactor is not decreased even if a tapered portion is provided, comparedwith the case where there is no tapered portion. FIG. 3 shows thatcombination regions with Q/Q₀ equal to or larger than 1.1, 1.2, and 1.3are present.

In addition, the reason why the Q factor decreases as the taper widthoccupancy ξ increases is presumably that flexural vibration of theresonating arm 30 moves the entire base portion 20 and causes largedistortion (temperature gradient) in a cut portion 55 (refer to FIG. 1).

Accordingly, by controlling the shape of the tapered portion 25 with thetaper length occupancy η and the taper width occupancy ξ, it is possibleto find out the shape of the tapered portion 25 capable of suppressingthe thermoelastic loss. As a result, it is possible to realize theresonator element 10 with a stable vibration mode by increasing the Qfactor compared with the case where the tapered portion 25 is notprovided or the case where a tapered portion is simply provided withouttaking the taper length occupancy η and the taper width occupancy ξ intoconsideration.

Moreover, as it is clear from FIG. 3 that an appropriate combination ofthe taper length occupancy η and the taper width occupancy ξ isnecessary in order to obtain more preferable Q/Q₀, specific exampleswill be described hereinafter.

FIRST EXAMPLE

First, a range where Q/Q₀ is equal to or larger than 1.1 will bedescribed with reference to FIG. 4.

FIG. 4 is a graph showing the relationship between the taper lengthoccupancy η (horizontal axis) at which Q/Q₀ is equal to or larger than1.1 and the taper width occupancy ξ (vertical axis). As the taper lengthoccupancy η and the taper width occupancy ξ used here, the rangesthereof where Q/Q₀ is equal to or larger than 1.1 are extracted fromFIG. 3 described above.

In FIG. 4, an uppermost curve shows an upper limit range of the taperwidth occupancy ξ with respect to a change of the taper length occupancyη at which Q/Q₀ is equal to or larger than 1.1, a lowermost curve showsa lower limit range, and a middle curve shows a graph of the case thatonly the maximum value (ξmax) are plotted. In the uppermost graph, thetaper width occupancy ξ is expressed asξ=−4.723×10¹η²+1.434×10¹η+1.711×10⁻² when the range of the taper lengthoccupancy η is 0.0168≦η≦0.1680.

Moreover, in the lowermost graph, the taper width occupancy ξ isexpressed as ξ=2.409η²+4.728×10⁻²η30 2.959×10⁻² when the range of thetaper length occupancy η is 0.0168≦η≦0.1680.

That is, when the range of the taper length occupancy η is0.0168≦η≦0.1680, the taper width occupancy at which Q/Q₀≧1.1 issatisfied is expressed by the following expression.2.409η²+4.728×10⁻²η+2.959×10⁻²≦ξ≦−4.723×10¹η²+1.434×10¹η+1.711×10⁻²  (1)

By setting the taper width occupancy ξ within the range of expression(1) with respect to the taper length occupancy η, it is possible toobtain the Q factor 1.1 times or more the Q factor when a taperedportion is not provided. As a result, the resonator element 10 with astable vibration mode can be realized.

SECOND EXAMPLE

Next, the range where Q/Q₀ is equal to or larger than 1.2 will bedescribed with reference to FIG. 5.

FIG. 5 is a graph showing the relationship between the taper lengthoccupancy η (horizontal axis) and the taper width occupancy ξ (verticalaxis) when Q/Q₀ is equal to or larger than 1.2. As the taper lengthoccupancy η and the taper width occupancy ξ used here, the rangesthereof where Q/Q₀ is equal to or larger than 1.2 are extracted fromFIG. 3 described above.

In FIG. 5, an uppermost curve shows an upper limit range of the taperwidth occupancy ξ with respect to a change of the taper length occupancyη at which Q/Q₀ is equal to or larger than 1.2, a lowermost curve showsa lower limit range, and a middle curve shows a graph of the case thatonly the maximum values (ξmax) are plotted. In the uppermost graph, thetaper width occupancy ξ is expressed asξ=−5.255×10¹η²+1.394×10¹η−1.219×10⁻¹ when the range of the taper lengthoccupancy η is 0.0168≦η≦0.1680.

Moreover, in the lowermost graph, the taper width occupancy ξ isexpressed as ξ=9.417η²−6.358×10⁻¹η+9.984×10⁻² when the range of thetaper length occupancy η is 0.0168≦η≦0.1680.

That is, when the range of the taper length occupancy η is0.0168≦η≦0.1680, the taper width occupancy ξ at which Q/Q₀≦1.2 issatisfied is expressed by the following expression.9.417η²−6.358×10⁻¹η+9.984×10⁻²≦ξ≦−5.255×10¹η²+1.394×10¹η−1.219×10⁻¹  (2)

By setting the taper width occupancy ξ within the range of expression(2) with respect to the taper length occupancy η, it is possible toobtain the Q factor 1.2 times or more compared to the Q factor when atapered portion is not provided. As a result, the resonator element 10with a more stable vibration mode can be realized.

THIRD EXAMPLE

Next, the range where Q/Q₀ is equal to or larger than 1.3 will bedescribed with reference to FIG. 6.

FIG. 6 is a graph showing the relationship between the taper lengthoccupancy η (horizontal axis) and the taper width occupancy ξ (verticalaxis) when Q/Q₀ is equal to or larger than 1.3. As the taper lengthoccupancy η and the taper width occupancy ξ used here, the rangesthereof where Q/Q₀ is equal to or larger than 1.3 are extracted fromFIG. 3 described above.

In FIG. 6, an uppermost curve shows an upper limit range of the taperwidth occupancy ξ with respect to a change of the taper length occupancyη, a lowermost curve shows a lower limit range, and a middle curve showsa graph of the case that only the maximum values (ξmax) are plotted. Inthe uppermost graph, the taper width occupancy ξ is expressed asξ=−9.160×10¹η²+1.899×10¹η−4.679×10⁻¹ when the range of the taper lengthoccupancy η is 0.0504≦η≦0.1260.

Moreover, in the lowermost graph, the taper width occupancy ξ isexpressed as ξ=5.191×10¹η²−6.959η+4.339×10⁻¹ when the range of the taperlength occupancy η is 0.0504≦η≦0.1260.

That is, when the range of the taper length occupancy η is0.0504≦η≦0.1260, the taper width occupancy ξ at which Q/Q₀≧1.3 issatisfied is expressed by the following expression.5.191×10¹η²−6.959η+4.339×10⁻¹≦ξ≦−9.160×10¹η²+1.899×10¹η−4.679×10⁻¹   (3)

By setting the taper width occupancy ξ within the range of expression(3) with respect to the taper length occupancy η, it is possible toobtain the Q factor of 1.3 times or more compared to the Q factor when atapered portion is not provided. As a result, the resonator element 10with a more stable vibration mode can be realized.

Note that, in the first to third examples described above, the taperedportion 25 is of a shape formed by one straight line connecting theposition of the tapered portion connected to the resonating arm 30 andthe position thereof connected to the base portion 20. However, if thetapered portion is of a shape formed by connecting a plurality ofstraight lines or a curved line, it is possible to further reduce thethermoelastic loss and to increase the Q factor. Specific examplesthereof will be described with reference to the accompanying drawings.

FOURTH EXAMPLE

Next, a fourth example will be described with reference to theaccompanying drawings. The fourth example is characterized in that thetapered portion 25 is formed by connection of a plurality of straightlines, while the tapered portion 25 is formed by one straight line inthe first to third examples. Now, the tapered portion 25 will bedescribed using an enlarged view thereof.

FIG. 7 is a partial plan view showing a tapered portion related to thefourth example. Note that, although the tapered portion 25 is formed intwo places for each of the two resonating arms 30, that is, in a totalof four places, only one of these places is shown in FIG. 7. As shown inFIG. 7, the tapered portion 25 is formed by connecting three straightlines. Here, it is assumed that the position of the tapered portionconnected to the resonating arm 30 is a P1 position, the positionthereof connected to the base portion 20 is a P4 position, and thepositions thereof at which the straight line is discontinuous betweenthe P1 and P4 positions are P2 and P3 positions.

In addition, a crossing angle formed by one straight line, whichconnects the position (P1 position) connected to the resonating arm 30and the position (P4 position) connected to the base portion 20, and theextended line of the resonating arm 30 is expressed as θ1, a crossingangle formed by the extended line of the straight line which connectsthe P1 and P2 positions to each other, and the straight line whichconnects the P2 and P3 positions to each other is expressed as θ2, and acrossing angle formed by the extended line of the straight line whichconnects the P2 and P3 positions to each other, and the straight linewhich connects the P3 and P4 positions to each other is expressed as θ3.The shape of the one straight line which connects the P1 and P4positions to each other is equivalent to the shape of the taperedportion 25 described in the first to third examples.

In the tapered portion 25 of the present example, the crossing angles θ2and θ3 in discontinuous portions (P2 and P3 positions) are smaller thanthe crossing angle θ1 between the extended line of the resonating arm 30and the straight line connecting the P1 and P4 positions to each other.In addition, the tapered portion 25 is located in a triangle formed bystraight lines connecting Lt in a range which satisfies the conditionsof the taper length occupancy η and the taper width occupancy ξ in thefirst to third examples described above, to Wtmax and Wtmin when the Ltis fixed.

Thus, if the tapered portion 25 is made to have a shape formed byconnecting a plurality of straight lines, stress generated in thetapered portion is dispersed compared with the case where the taperedportion 25 is formed by one straight line. Accordingly, distortion isalso dispersed, temperature gradient is decreased and thus the heat flowis suppressed. As a result, a thermoelastic loss is reduced, and the Qfactor can be increased.

Therefore, as the number of straight lines discontinuously connected toeach other increases (that is, as the number of discontinuous portionsincreases) in the tapered portion 25, stress generated in the taperedportion is dispersed, so that a thermoelastic loss can be reduced.

FIFTH EXAMPLE

Next, a fifth example will be described with reference to theaccompanying drawings. The fifth example is characterized in that thetapered portion 25 is formed by a curved line, while the tapered portion25 is formed by connecting a plurality of straight lines to each otherin the fourth example described above. Now, the tapered portion 25 willbe described using an enlarged view thereof.

FIG. 8 is a partial plan view showing a tapered portion related to thefifth example. Note that, although the tapered portion 25 is formed intwo places for each of the two resonating arms 30, that is, in a totalof four places, only one of these places is shown in FIG. 8. As shown inFIG. 8, the tapered portion 25 is formed by one curve connecting theposition (P1 position) which is connected to the resonating arm 30, andthe position (P4 position) which is connected to the base portion 20.

Here, the crossing angle between the curve and the extended line of theresonating arm 30 is set to θ4. That is, the crossing angle θ4 is anangle formed by the tangential line of the curve of the tapered portion25 at the P1 position and the extended line of the resonating arm 30. Inaddition, the crossing angle formed by the extended line of theresonating arm and the straight line which connects the position (P1position) connected to the resonating arm 30 and the position (P4position) connected to the base portion 20, is set to θ5. In addition,the crossing angle θ4 is set to be smaller than the crossing angle θ5.

In addition, the tapered portion 25 is located in a triangle formed bystraight lines connecting Lt in a range which satisfies the conditionsof the taper length occupancy η and the taper width occupancy ξ in thefirst to third examples described above, to Wtmax and Wtmin when the Ltis fixed.

As previously described in the fourth example, the stress generated inthe tapered portion is dispersed and the thermoelastic loss can bereduced as the number of straight lines discontinuously connected toeach other increases in the tapered portion 25. Accordingly, if thenumber of straight lines discontinuously connected to each other is madeinfinite, the tapered portion 25 becomes a curve. Thus, by forming thetapered portion 25 as a curved line, the stress generated in the taperedportion is dispersed compared with the case where the tapered portion 25is formed by using a straight line. Accordingly, distortion is alsodispersed, temperature gradient is decreased and the heat flow issuppressed. As a result, a thermoelastic loss is further reduced, andthe Q factor can be increased.

Note that, if the conditions of each of the first to fourth examplesdescribed above are satisfied, the same effects can be acquired even ifthe position of the tapered portion 25 is changed. This will bedescribed in a sixth example.

SIXTH EXAMPLE

The sixth example is characterized in that the position of the taperedportion 25 connected to the base portion 20 is located in an innerposition from the +X-direction end of the base portion 20, while theconnection position of the tapered portion 25 is at the +X-direction endof the base portion 20 in the first to fifth examples described above.Now, the tapered portion 25 will be described using an enlarged viewthereof.

FIG. 9 is a partial plan view showing a tapered portion related to thesixth example. Note that, although the tapered portion 25 is formed intwo places for each of the two resonating arms 30, that is, in a totalof four places, only one of these places is shown in FIG. 9. As shown inFIG. 9, the tapered portion 25 is formed by one straight line connectingthe position (P1 position) which is connected to the resonating arm 30,and the position (P4 position) which is connected to the base portion20.

Here, the P4 position is located in an inner position from the+X-direction end of the base portion 20, and a straight portion 21 isformed between the P4 position and the +X-direction end.

Note that, the tapered portion 25 may have a shape obtained byconnecting a plurality of straight lines as in the fourth example or maybe of a shape formed by one continuous curved line as in the fifthexample. Even if the tapered portion 25 has such shapes, the sameeffects as in the examples described above can be acquired.

In addition, although the tuning fork type resonator element has beendescribed as an example of the resonator element 10, it may also beapplied to an acceleration sensor or a gyro sensor.

Although the spindle portion 40 is provided at the distal end of theresonating arm 30, the spindle portion 40 may not be provided. When thespindle portion is provided, it is more preferable that a taperedportion based on the technical idea of each example described above isprovided in the connection portion between the spindle portion and theresonating arm.

In addition, the support arm 50 may be removed, and a fixed portion ofthe resonator element 10 may be made to serve as the base portion 20.

Resonator

Next, an example of a resonator using the resonator element 10 will bedescribed with reference to the accompanying drawings.

FIGS. 10A and 10B are views showing the schematic configuration of aresonator. FIG. 10A is a plan view, and FIG. 10B is a cross-sectionalview taken along the line E-E of FIG. 10A. In FIGS. 10A and 10B, aresonator 70 is configured to include the resonator element 10 and apackage 80 in which the resonator element 10 is housed.

The package 80 includes a package base 81, a seam ring 82, a lid 85, andthe like. In the package base 81, a recess is formed so that theresonator element 10 can be accommodated. In the recess, a connectionpad 88 connected to amount electrode (not shown) of the resonatorelement 10 is provided. The connection pad 88 is connected to a wiringline in the package base 81 and is formed so as to be electricallyconnective to an external connection terminal 83 provided in the outerperiphery of the package base 81.

The seam ring 82 is provided around the recess of the package base 81.In addition, a through hole 86 is provided in a bottom portion of thepackage base 81.

The resonator element 10 is bonded and fixed to the connection pad 88 ofthe package base 81 with a conductive adhesive 84 by the support arm 50.Moreover, in the package 80, the lid 85 which covers the recess of thepackage base 81 and the seam ring 82 are seam-welded.

A sealant 87 formed of a metallic material or the like is filled in thethrough hole 86 of the package base 81. The sealant 87 is solidifiedafter being melted in a decompressed atmosphere to airtightly seal thethrough hole 86, so that the inside of the package base 81 can maintaina decompressed state or a vacuum state.

The resonator element 10 is excited by a driving signal supplied fromthe outside through an external connection terminal 83 and as a result,the resonator 70 oscillates (resonates) at a predetermined frequency(for example, 32 kHz).

As described above, since the resonator 70 has the resonator element 10in which the tapered portion described in the first to sixth examples isformed, the Q factor can be increased by suppressing vibration leakage.As a result, stable vibration characteristics can be obtained. Inaddition, it is possible to prevent the resonator element 10 from beingbroken by an impact or prevent deterioration of a vibration mode.

In addition, since the resonator element 10 is housed in the package 80and the resonator element 10 is vibrated in a vacuum environment withinthe package 80, it is possible to maintain more stable vibration over along period of time. In addition, since the resonator element 10 ishoused in the package, it is easy in handling and it can be protectedagainst external environments, such as humidity.

What is claimed is:
 1. A vibrating piece comprising: a base; and avibrating arm that has a tapered portion extending from the base andthat has an arm portion extending from the tapered portion, thevibrating arm performing flexural vibration, wherein a width of thetapered portion increases toward the base, and when a length of thevibrating arm is L, a width of the arm portion is W, a length of thetapered portion along with the arm portion is Lt, a width of the taperedportion along with the base is Wt, a taper length occupancy is η=Lt/L,and a taper width occupancy is ξ=2Wt/W, the following formulas aresatisfied:0.0168≦η≦0.1680;2.409η²+4.728×10⁻²η+2.959×10⁻²≦ξ≦−4.723×10¹η²+1.434×10¹η+1.711×10⁻²; andwherein L≠1.65 mm, W≠0.11 mm, Lt≠0.2 mm, and Wt≠0.01 mm.
 2. Thevibrating piece according to claim 1, wherein an outer circumference ofthe tapered portion is one of a straight line or a curved line.
 3. Thevibrating piece according to claim 2, wherein the outer circumference ofthe tapered portion is formed between a first position that connectsbetween the arm portion and a first end of the outer circumference and asecond position that connects between the base and a second end of theouter circumference, the outer circumference is formed by connection ofa plurality of straight lines, and crossing angles between the pluralityof straight lines are smaller than a crossing angle between an extendedline of the vibrating arm and a straight line connecting the first andsecond positions.
 4. The vibrating piece according to claim 2, whereinthe outer circumference of the tapered portion is formed between a firstposition that connects between the arm portion and a first end of theouter circumference and a second position that connects between the baseand a second end of the outer circumference, the outer circumference isformed by one continuous curved line connecting the first and secondpositions, and a crossing angle between the curved line and an extendedline of the vibrating arm is smaller than a crossing angle between theextended line of the vibrating arm and a straight line connecting thefirst and second positions.
 5. The resonator element according to claim1, wherein the vibrating arm has a weight provided at a tip of the armportion.
 6. The resonator element according to claim 5, wherein a widthof the weight is larger than the width of the arm portion.
 7. A vibratorcomprising: a vibrating piece according to claim 1; and a package thathouses the vibrating piece.
 8. A vibrating piece comprising: a base; anda vibrating arm that has a tapered portion extending from the base andthat has an arm portion extending from the tapered portion, thevibrating arm performing flexural vibration, wherein a width of thetapered portion increases toward the base, and when a length of thevibrating arm is L, a width of the arm portion is W, a length of thetapered portion along with the arm portion is Lt, a width of the taperedportion along with the base is Wt, a taper length occupancy is η=Lt/L,and a taper width occupancy is ξ=2Wt/W, the following formulas aresatisfied:0.0168≦η≦0.1680;9.417η²−6.358×10⁻¹η+9.984×10⁻²≦ξ≦−5.255×10¹η²+1.394×10¹η−1.219×10⁻¹; andwherein L≠1.65 mm, W≠0.11 mm, Lt≠0.2 mm, and Wt≠0.01 mm.
 9. Thevibrating piece according to claim 8, wherein an outer circumference ofthe tapered portion is one of a straight line or a curved line.
 10. Thevibrating piece according to claim 9, wherein the outer circumference ofthe tapered portion is formed between a first position that connectsbetween the arm portion and a first end of the outer circumference and asecond position that connects between the base and a second end of theouter circumference, the outer circumference is formed by connection ofa plurality of straight lines, and crossing angles between the pluralityof straight lines are smaller than a crossing angle between an extendedline of the vibrating arm and a straight line connecting the first andsecond positions.
 11. The vibrating piece according to claim 9, whereinthe outer circumference of the tapered portion is formed between a firstposition that connects between the arm portion and a first end of theouter circumference and a second position that connects between the baseand a second end of the outer circumference, the outer circumference isformed by one continuous curved line connecting the first and secondpositions, and a crossing angle between the curved line and an extendedline of the vibrating arm is smaller than a crossing angle between theextended line of the vibrating arm and a straight line connecting thefirst and second positions.
 12. The resonator element according to claim8, wherein the vibrating arm has a weight provided at a tip of the armportion.
 13. The resonator element according to claim 12, wherein awidth of the weight is larger than the width of the arm portion.
 14. Avibrator comprising: a vibrating piece according to claim 8; and apackage that houses the vibrating piece.